Inverse Inequality Estimates with Symbolic Computation
Christoph Koutschan, Martin Neumüller, Silviu Radu
AbstractIn the convergence analysis of numerical methods for solving partial differential equations (such as finite element methods) one arrives at certain generalized eigenvalue problems, whose maximal eigenvalues need to be estimated as accurate as possible. We apply symbolic computation methods to the situation of square elements and are able to improve the previously known upper bound by a factor of 8. More precisely, we try to evaluate the corresponding determinant using the holonomic ansatz, which is a powerful tool for dealing with determinants, proposed by Zeilberger in 2007. However, it turns out that this method does not succeed on the problem at hand. As a solution we present a variation of the original holonomic ansatz that is applicable to a larger class of determinants, including the one we are dealing with here. We obtain an explicit closed form for the determinant, whose special form enables us to derive new and tight upper resp. lower bounds on the maximal eigenvalue, as well as its asymptotic behaviour.
PaperRead the full paper here.
Supplementary electronic material
- MaximalEigenvalueProblem.nb (Mathematica notebook)
- MaximalEigenvalueProblem.cdf (the notebook in cdf format for the Wolfram CDF Player)
- MaximalEigenvalueProblem.pdf (pdf screenshot of the notebook)
- For executing the commands in the notebook, the additional Mathematica packages HolonomicFunctions and Guess are required.
- For the proof of Lemma 3.3 the two files cert0.m and cert1.m need to be read into Mathematica. They contain certificates for the summation with respect to j.